How to Find Highest Common Factor (HCF)? Complete Method, Examples, Notes & Worksheet

Introduction

The highest common factor (HCF) is an important concept in mathematics. It is used to find the greatest number that can divide two or more numbers exactly without leaving a remainder.

In this article, you will learn how to find the highest common factor, different methods of finding HCF, solved examples, short notes, MCQs, and a practice worksheet.

These notes are especially useful for students who want to understand the highest common factor method in a simple and step-by-step way.

What is Highest Common Factor?

The Highest Common Factor (HCF) of two or more numbers is the greatest number that divides all of the given numbers exactly.

It is also called the Greatest Common Factor (GCF).

Example

Find the HCF of 12 and 18.

Factors of 12:

1, 2, 3, 4, 6, 12

Factors of 18:

1, 2, 3, 6, 9, 18

Common factors are:

1, 2, 3, 6

The highest common factor is 6.

Therefore:

HCF of 12 and 18 = 6

How to Calculate Highest Common Factor?

There are several methods for finding the HCF of numbers. The most commonly used methods are:

1. Listing Factors Method
2. Prime Factorization Method
3. Division Method

Let us understand each method.

Method 1: Listing Factors Method

In this method, we list all the factors of each number. Then we identify the common factors and select the greatest one.

Example 1

Find the HCF of 24 and 36.

Step 1: Find the factors of 24

Factors of 24 are:

1, 2, 3, 4, 6, 8, 12, 24

Step 2: Find the factors of 36

Factors of 36 are:

1, 2, 3, 4, 6, 9, 12, 18, 36

Step 3: Find common factors

Common factors are:

1, 2, 3, 4, 6, 12

Step 4: Select the highest common factor

The greatest common factor is 12.

Therefore:

HCF = 12

Method 2: Prime Factorization Method

In the prime factorization method, we express each number as a product of prime numbers.

Then we identify the prime factors common to all the numbers and multiply them together.

Example 2

Find the HCF of 24 and 36 using prime factorization.

Prime factorization of 24:

24 = 2 × 2 × 2 × 3

Prime factorization of 36:

36 = 2 × 2 × 3 × 3

Common prime factors are:

2 × 2 × 3

Therefore:

HCF = 2 × 2 × 3

HCF = 12

Method 3: Division Method

The division method is also called the Euclidean algorithm. It is particularly useful when the numbers are large.

Example 3

Find the HCF of 48 and 18.

Divide the larger number by the smaller number:

48 ÷ 18 = 2 remainder 12

Now divide 18 by 12:

18 ÷ 12 = 1 remainder 6

Now divide 12 by 6:

12 ÷ 6 = 2 remainder 0

When the remainder becomes zero, the last non-zero remainder is the HCF.

Therefore:

HCF of 48 and 18 = 6

Highest Common Factor Method Using Prime Factors

Another convenient way to understand HCF is by using prime factors.

Suppose we need to find the HCF of 60 and 90.

Prime factorization of 60:

60 = 2 × 2 × 3 × 5

Prime factorization of 90:

90 = 2 × 3 × 3 × 5

The common prime factors are:

2 × 3 × 5

Therefore:

HCF = 2 × 3 × 5

HCF = 30

How to Find HCF of Three Numbers?

The same methods can be used when there are three or more numbers.

Example 4

Find the HCF of 18, 24 and 30.

Prime factorization:

18 = 2 × 3 × 3

24 = 2 × 2 × 2 × 3

30 = 2 × 3 × 5

The prime factors common to all three numbers are:

2 × 3

Therefore:

HCF = 6

HCF and Factors

A factor is a number that divides another number exactly.

For example, the factors of 20 are:

1, 2, 4, 5, 10, 20

If a number is a factor of two or more numbers, it is called a common factor.

The greatest of these common factors is called the highest common factor.

Example

For 16 and 24:

Factors of 16:

1, 2, 4, 8, 16

Factors of 24:

1, 2, 3, 4, 6, 8, 12, 24

Common factors:

1, 2, 4, 8

Therefore:

HCF = 8

Important Points About HCF

• HCF stands for Highest Common Factor.
• HCF is the greatest number that divides all given numbers exactly.
• HCF can be found by listing factors.
• HCF can be found by prime factorization.
• HCF can also be found using the division method.
• The HCF of two or more numbers cannot be greater than the smallest number.
• If one number is a factor of another number, the smaller number is the HCF.
• The HCF of two different prime numbers is usually 1.
• The HCF of two consecutive positive integers is always 1.

Solved Examples

Example 5

Find the HCF of 15 and 25.

Factors of 15:

1, 3, 5, 15

Factors of 25:

1, 5, 25

Common factors:

1, 5

Therefore:

HCF = 5

Example 6

Find the HCF of 32 and 48.

Prime factorization:

32 = 2 × 2 × 2 × 2 × 2

48 = 2 × 2 × 2 × 2 × 3

Common factors:

2 × 2 × 2 × 2

Therefore:

HCF = 16

Example 7

Find the HCF of 45 and 75.

Prime factorization:

45 = 3 × 3 × 5

75 = 3 × 5 × 5

Common prime factors:

3 × 5

Therefore:

HCF = 15

Short Notes on Highest Common Factor

Definition

The highest common factor of two or more numbers is the greatest number that divides all the given numbers exactly.

Methods of Finding HCF

The main methods are:

1. Listing Factors Method – List the factors of each number and select the greatest common factor.
2. Prime Factorization Method – Write numbers as products of prime factors and multiply the common prime factors.
3. Division Method – Use repeated division until the remainder becomes zero. The last non-zero remainder is the HCF.

Example

Find the HCF of 20 and 30.

Factors of 20:

1, 2, 4, 5, 10, 20

Factors of 30:

1, 2, 3, 5, 6, 10, 15, 30

Common factors:

1, 2, 5, 10

Therefore:

HCF = 10

MCQs on Highest Common Factor

1. What does HCF stand for?

A. Highest Common Fraction
B. Highest Common Factor
C. Highest Calculated Factor
D. Highest Common Formula

Answer: B. Highest Common Factor

2. What is the HCF of 12 and 18?

A. 2
B. 3
C. 6
D. 9

Answer: C. 6

3. What is the HCF of 20 and 30?

A. 5
B. 10
C. 15
D. 20

Answer: B. 10

4. What is the HCF of 16 and 24?

A. 4
B. 6
C. 8
D. 12

Answer: C. 8

5. Which method uses prime numbers to find HCF?

A. Prime Factorization Method
B. Addition Method
C. Subtraction Method
D. Average Method

Answer: A. Prime Factorization Method

6. What is the HCF of 15 and 25?

A. 3
B. 5
C. 10
D. 15

Answer: B. 5

7. What is the HCF of 18, 24 and 30?

A. 3
B. 4
C. 6
D. 12

Answer: C. 6

8. What is the HCF of two different prime numbers?

A. 0
B. 1
C. 2
D. 10

Answer: B. 1

9. In the division method, when the remainder becomes zero, the HCF is:

A. The first quotient
B. The last quotient
C. The last non-zero remainder
D. The first remainder

Answer: C. The last non-zero remainder

10. What is the HCF of 32 and 48?

A. 8
B. 12
C. 16
D. 24

Answer: C. 16

Highest Common Factor Worksheet / Assignment

Part A: Find the HCF by Listing Factors

1. Find the HCF of 8 and 12.
2. Find the HCF of 18 and 27.
3. Find the HCF of 20 and 28.
4. Find the HCF of 24 and 32.
5. Find the HCF of 30 and 45.

Part B: Find the HCF by Prime Factorization

6. Find the HCF of 36 and 48.
7. Find the HCF of 40 and 60.
8. Find the HCF of 42 and 56.
9. Find the HCF of 54 and 72.
10. Find the HCF of 60 and 90.

Part C: Find the HCF of Three Numbers

11. Find the HCF of 12, 18 and 24.
12. Find the HCF of 15, 20 and 25.
13. Find the HCF of 18, 27 and 36.
14. Find the HCF of 24, 36 and 48.
15. Find the HCF of 30, 45 and 60.

Part D: Challenge Questions

16. Find the HCF of 84 and 126.
17. Find the HCF of 96 and 144.
18. Find the HCF of 72, 108 and 144.
19. Find the HCF of 120 and 180.
20. Find the HCF of 150, 225 and 300.

Answers to Worksheet

1. 4
2. 9
3. 4
4. 8
5. 15
6. 12
7. 20
8. 14
9. 18
10. 30
11. 6
12. 5
13. 9
14. 12
15. 15
16. 42
17. 48
18. 36
19. 60
20. 75

Frequently Asked Questions About Highest Common Factor

What is the highest common factor?

The highest common factor is the greatest number that divides two or more given numbers exactly.

How do you calculate the highest common factor?

You can calculate the HCF by listing factors, using prime factorization, or applying the division method.

What is the easiest method to find HCF?

For small numbers, listing factors is usually simple. For larger numbers, prime factorization or the division method can be more convenient.

Is HCF the same as GCF?

Yes. HCF (Highest Common Factor) and GCF (Greatest Common Factor) refer to the same mathematical concept.

What is the HCF of two prime numbers?

The HCF of two different prime numbers is 1 because they have no common factor other than 1.

Conclusion

The highest common factor is the greatest common factor of two or more numbers. It can be found using the listing factors method, prime factorization method, or division method.

Students should practise all three methods so that they can choose the most suitable method according to the numbers given in a question.

For more Class 9 Mathematics learning material, students can explore Nisar Math Academy for recorded lectures, notes, assessments, lesson plans, and other useful educational resources. Free learning resources are also available, while the complete course can be accessed through website membership.

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